Parametrizations of toric varieties over any field

The columns of an integral matrix D give rise to the toric variety VK(ID) and also provide a parametrization of a subset of VK(ID), the so-called toric set ΓK(D). We completely determine the toric set ΓK(D) over any field. We provide conditions under which VK(ID) is fully parametrized by the columns of D, that means ΓK(D)=VK(ID). In particular, we prove that normal toric varieties over any field are always fully parametrized by the columns of an appropriate matrix.